This note advances knowledge of the threshold of prox-boundedness of a function; an important concern in the use of proximal point optimization algorithms and in determining the existence of the Moreau envelope of the function. In finite dimensions, we study general prox-bounded functions and then focus on some useful classes such as piecewise functions and Lipschitz continuous functions. The thresholds are explicitly determined when possible and bounds are established otherwise. Some calculus rules are constructed; we consider functions with known thresholds and find the thresholds of their sum and composition.

中文翻译：

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接近极限阈值的存在，标识和演算规则的条件

本说明提高了对函数的有界边界阈值的了解；在使用近端优化算法和确定函数的Moreau包络的存在中，是一个重要的问题。在有限维中，我们研究通用的有边界约束的函数，然后重点研究一些有用的类，例如分段函数和Lipschitz连续函数。在可能的情况下明确确定阈值，否则确定界限。构造了一些微积分规则；我们考虑具有已知阈值的函数，并找到它们的和与组成的阈值。